A Viscoelastic Fictitious Crack Model for the Fracture of Sea Ice

The fracture of sea ice is modeled using a viscoelastic fictitious crack(cohesive zone) model. The sea ice is modeled as a linear viscoelastic material. The fictitious crack model is implemented via the weight function method. The associated stress-separation curve can be rate dependent. The impact of assuming viscoelastic behavior in the bulk as opposed to elastic behavior is studied. Results from the model are compared to the available exact results for various test cases. The model is applied to a large scale in situ sea ice fracture test. Various implications of such applications are pointed out. This viscoelastic fictitious crack model is found to be a promising tool in investigations pertaining to the fracture of sea ice. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical study of geometric constraint and cohesive parameters in steady-state viscoelastic crack growth

This paper presents a finite element study of cohesive crack growth in a thin infinite viscoelastic strip to investigate the effects of viscoelastic properties, strip height, and cohesive model parameters on the crack growth resistance. The results of the study show that the dependence of the fracture energy on the viscoelastic properties for the strip problem is similar to that obtained for the infinite body problem even when the cohesive zone length is large compared to the height of the strip. The fracture energy also depends on the crack speed v through the dimensionless parameter v τ/L _∞ where L _∞ is the characteristic length of the cohesive zone and τ is the characteristic relaxation time of the bulk material. This relationship confirms that at least two properties of the fracture process must be prescribed accurately to model viscoelastic crack growth. In contrast, the fracture energy and crack speed are insensitive to the strip height even in situations where the growth of the dissipation zone is severely constrained by the strip boundaries. We observe that at high speeds, where the fracture energy asymptotically approaches the maximum value, the material surrounding the cohesive zone is in the rubbery (equilibrium) state and not the glassy state.     

A cohesive finite element formulation for modelling fracture and delamination in solids

In recent years, cohesive zone models have been employed to simulate fracture and delamination in solids. This paper presents in detail the formulation for incorporating cohesive zone models within the framework of a large deformation finite element procedure. A special Ritz-finite element technique is employed to control nodal instabilities that may arise when the cohesive elements experience material softening and lose their stress carrying capacity. A few simple problems are presented to validate the implementation of the cohesive element formulation and to demonstrate the robustness of the Ritz solution method. Finally, quasi-static crack growth along the interface in an adhesively bonded system is simulated employing the cohesive zone model. The crack growth resistance curves obtained from the simulations show trends similar to those observed in experimental studies     

Asymptotic stress intensity factor density profiles for smeared-tip method for cohesive fracture

The paper presents a computational approach and numerical data which facilitate the use of the smeared-tip method for cohesive fracture in large enough structures. In the recently developed K-version of the smeared tip method, the large-size asymptotic profile of the stress intensity factor density along a cohesive crack is considered as a material characteristic, which is uniquely related to the softening stress-displacement law of the cohesive crack. After reviewing the K-version, an accurate and efficient numerical algorithm for the computation of this asymptotic profile is presented. The algorithm is based on solving a singular Abel's integral equation. The profiles corresponding to various typical softening stress-displacement laws of the cohesive crack model are computed, tabulated and plotted. The profiles for a certain range of other typical softening laws can be approximately obtained by interpolation from the tables. Knowing the profile, one can obtain with the smeared-tip method an analytical expression for the large-size solution to fracture problems, including the first two asymptotic terms of the size effect law. Consequently, numerical solutions of the integral equations of the cohesive crack model as well as finite element simulations of the cohesive crack are made superfluous. However, when the fracture process zone is attached to a notch or to the body surface and the cohesive zone ends with a stress jump, the solution is expected to be accurate only for large-enough structures.     

Cohesive Crack Model with Rate-Dependent Opening and Viscoelasticity: II. Numerical Algorithm, Behavior and Size Effect

In the preceding companion paper (Bažant and Li, 1995), the solution of an aging viscoelastic law was structure containing a cohesive crack with a rate-dependent stress-displacement softening law was reduced to a system of one-dimensional integro-differential equations involving compliance functions for points on the crack faces and the load point. An effective numerical algorithm for solving these equations, which dramatically reduces the computer time compared to the general two-dimensional finite element solution, is presented. The behavior of the model for various loading conditions is studied. It is shown that the model can closely reproduce the available experimental data from fracture tests with different loading rates spanning several orders of magnitude, and tests with sudden changes of the loading rate. Influence of the loading rate on the size effect and brittleness is also analyzed and is shown to agree with experiments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Damage Dependent Constitutive Behavior and Energy Release Rate for a Cohesive Zone in a Thermoviscoelastic Solid

In this paper a cohesive zone is introduced ahead of a crack tip in order to avoid the singularity at the crack tip. By applying thermodynamics to the cohesive zone and the surrounding body, a fracture criterion will be established so that the inelastic energy dissipation both in the cohesive zone and the surrounding bulk material can be distinguished from the energy released by fracture, and the propagation of crack can be predicted. In addition, the cohesive zone constitutive equation is constructed utilizing the Helmholtz free energy in the form of a single hereditary integral for a nonlinear viscoelastic material. The resulting constitutive model for the cohesive zone contains an internal state variable which represents the damage state within the cohesive zone. When the cohesive zone opening displacement is known, the energy release rate is thus history dependent, which is expressed in terms of the damage state, the length of separation in the cohesive zone and the geometric configuration of the cohesive zone opening displacement. Example results contained herein demonstrate this effect. This revised version was published online in July 2006 with corrections to the Cover Date.     

A framework for fracture modelling based on the material forces concept with XFEM kinematics

A theoretical and computational framework which covers both linear and non‐linear fracture behaviour is presented. As a basis for the formulation, we use the material forces concept due to the close relation between on one hand the Eshelby energy–momentum tensor and on the other hand material defects like cracks and material inhomogeneities. By separating the discontinuous displacement from the continuous counterpart in line with the eXtended finite element method (XFEM), we are able to formulate the weak equilibrium in two coupled problems representing the total deformation. However, in contrast to standard XFEM, where the direct motion discontinuity is used to model the crack, we rather formulate an inverse motion discontinuity to model crack development. The resulting formulation thus couples the continuous direct motion to the inverse discontinuous motion, which may be used to simulate linear as well as non‐linear fracture in one and the same formulation. In fact, the linear fracture formulation can be retrieved from the non‐linear cohesive zone formulation simply by confining the cohesive zone to the crack tip. These features are clarified in the two numerical examples which conclude the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

黏聚裂纹阻抗的弯曲梁承载力

在混凝土类软化材料断裂研究中,裂纹端部损伤区被简化为具有黏聚应力分布的非线性裂纹,该黏 聚力对裂纹扩展有阻抗作用。裂纹体的应力强度因子是断裂力学标志载荷作用与几何构型因素的量化表达指标; 黏聚力形成的阻抗强度因子数值,与黏聚裂纹长度和材料极值拉伸应力存在数量关系。通过双K断裂判据,以 带切口的三点弯曲梁为断裂力学模型,分析了裂纹黏聚阻力对断裂过程的影响规律,计算该弯曲梁结构断裂试 样的最大承担载荷;其结果比较符合实验数据。     

GENERALIZED SIZE EFFECT EQUATION FOR QUASIBRITTLE MATERIALS

Abstract— Structures made of quasibrittle materials show a marked decrease in strength as their size increases; this is the well known size effect on strength. This contribution proposes a general formula to predict with great accuracy the size effect of pre-cracked specimens. Such a formula provides the basis to determine experimentally the fracture parameters of a cohesive crack model from the measurement of the peak load in specimens with an initial crack. The formula is applicable whenever the fracture of the material can be described by a cohesive crack model with initial linear softening, as is generally accepted for concrete. Experimental evidence is presented showing that the formula is reasonably accurate also for rock.     

Crack Growth Across a Strength Mismatched Bimaterial Interface

Crack growth across an interface between materials with different strength is examined by a cohesive zone model. The two materials have identical elastic properties but different fracture process properties, or different yield stresses, which is modeled by different cohesive stresses. The fracture criteria is a critical crack opening displacement. Load is represented by a stress intensity factor defining a remote square root singular stress field. The results show that the ratio between the cohesive stresses of the two materials primarily determines the behavior of the critical stress intensity factor. When the crack approaches a material with a higher cohesive stress the crack tip is shielded, but if the crack approaches a material with smaller critical crack opening displacement the maximum level of shielding is determined by the ratio between the critical crack opening displacements. When a crack approaches a material with a lower cohesive stress it is exposed to an amplified load. This revised version was published online in August 2006 with corrections to the Cover Date.     

Theory and numerics for finite deformation fracture modelling using strong discontinuities

A general finite element approach for the modelling of fracture is presented for the geometrically non‐linear case. The kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements. Thus, the deformation map is defined in terms of one continuous and one discontinuous portion, considered as mutually independent, giving rise to a weak formulation of the equilibrium consisting of two coupled equations. In addition, two different fracture criteria are considered. Firstly, a principle stress criterion in terms of the material Mandel stress in conjunction with a material cohesive zone law, relating the cohesive Mandel traction to a material displacement ‘jump’ associated with the direct discontinuity. Secondly, a criterion of Griffith type is formulated in terms of the material‐crack‐driving force (MCDF) with the crack propagation direction determined by the direction of the force, corresponding to the direction of maximum energy release. Apart from the material modelling, the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria. Copyright © 2005 John Wiley & Sons, Ltd.     

Cohesive crack analysis of size effect

This paper deals with the analysis of size effect in concrete. An extensive campaign of accurate numerical simulations, based on the cohesive crack model, is performed to compute the size effect curves (CSEC) for typical test configurations. The results are analyzed with reference to the classical Ba?ant’s size effect law (SEL) to investigate the relationship between CSEC and SEL. This analysis shows that as specimen size tends to infinity, the SEL represents the asymptote of the CSEC, and that the SEL parameter known as the effective fracture process zone length is a material property which can be expressed as a function of the cohesive crack law (CCL) parameters. Finally, the practical implications of this study are discussed in relation to the use of the CSEC or the SEL for the identification of the CCL parameters through the size effect method.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

On the Fracture Models Determined by the Continuum-strong Discontinuity Approach

The paper focuses on the Continuum Strong Discontinuity Approach (CSDA) to fracture mechanics, and the traction-separation cohesive laws induced from continuum dissipative models as their projections onto the failure interface. They are compared with the cohesive laws commonly used for the fracture simulation in quasi-brittle materials, typically concrete. Emphasis is placed in the analysis of the mechanical stress-strain states induced by the CSDA into the fracture process zone: first when the damage mechanism is initiated and, after, when the cohesive model determines the crack response. The influence of the material parameters, particularly the fracture energy and the initial continuum softening modulus, in the obtained phenomenological responses is also analyzed. Representative numerical solutions of fracture problems are finally presented.     

A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials

Numerical investigations are conducted to simulate high-speed crack propagation in pre-strained PMMA plates. In the simulations, the dynamic material separation is explicitly modeled by cohesive elements incorporating an initially rigid, linear-decaying cohesive law. Initial attempts using a rate-independent cohesive law failed to reproduce available experimental results as numerical crack velocities consistently overestimate experimental observations. As proof of concept, a phenomenological rate-dependent cohesive law, which bases itself on the physics of microcracking, is introduced to modulate the cohesive law with the macroscopic crack velocity. We then generalize this phenomenological approach by establishing a rate-dependent cohesive law, which relates the traction to the effective displacement and rate of change of effective displacement. It is shown that this new model produces numerical results in good agreement with experimental data. The analysis demonstrates that the simulation of high-speed crack propagation in brittle structures necessitates the use of rate-dependent cohesive models, which account for the complicated rate-process of dynamic fracture at the propagating crack tip.     

Cohesive models for damage evolution in laminated composites

A trend in the last decade towards models in which nonlinear crack tip processes are represented explicitly, rather than being assigned to a point process at the crack tip (as in linear elastic fracture mechanics), is reviewed by a survey of the literature. A good compromise between computational efficiency and physical reality seems to be the cohesive zone formulation, which collapses the effect of the nonlinear crack process zone onto a surface of displacement discontinuity (generalized crack). Damage mechanisms that can be represented by cohesive models include delamination of plies, large splitting (shear) cracks within plies, multiple matrix cracking within plies, fiber rupture or microbuckling (kink band formation), friction acting between delaminated plies, process zones at crack tips representing crazing or other nonlinearity, and large scale bridging by through-thickness reinforcement or oblique crack-bridging fibers. The power of the technique is illustrated here for delamination and splitting cracks in laminates. A cohesive element is presented for simulating three-dimensional, mode-dependent process zones. An essential feature of the formulation is that the delamination crack shape can follow its natural evolution, according to the evolving mode conditions calculated within the simulation. But in numerical work, care must be taken that element sizes are defined consistently with the characteristic lengths of cohesive zones that are implied by the chosen cohesive laws. Qualitatively successful applications are reported to some practical problems in composite engineering, which cannot be adequately analyzed by conventional tools such as linear elastic fracture mechanics and the virtual crack closure technique. The simulations successfully reproduce experimentally measured crack shapes that have been reported in the literature over a decade ago, but have not been reproduced by prior models.     

A discrete cohesive model for fractal cracks

The fractal crack model described here incorporates the essential features of the fractal view of fracture, the basic concepts of the LEFM model, the concepts contained within the Barenblatt-Dugdale cohesive crack model and the quantized (discrete or finite) fracture mechanics assumptions proposed by Pugno and Ruoff [Pugno N, Ruoff RS. Quantized fracture mechanics. Philos Mag 2004;84(27):2829-45] and extended by Wnuk and Yavari [Wnuk MP, Yavari A. Discrete fractal fracture mechanics. Engng Fract Mech 2008;75(5):1127-42]. The well-known entities such as the stress intensity factor and the Barenblatt cohesion modulus, which is a measure of material toughness, have been re-defined to accommodate the fractal view of fracture.For very small cracks or as the degree of fractality increases, the characteristic length constant, related to the size of the cohesive zone is shown to substantially increase compared to the conventional solutions obtained from the cohesive crack model. In order to understand fracture occurring in real materials, whether brittle or ductile, it seems necessary to account for the enhancement of fracture energy, and therefore of material toughness, due to fractal and discrete nature of crack growth. These two features of any real material appear to be inherent defense mechanisms provided by Nature.     

A modified cohesive zone model for a high‐speed expanding crack

The present work studies a self‐similar high‐speed expanding crack of mode‐I in a ductile material with a modified cohesive zone model. Compared with existing Dugdale models for moving crack, the new features of the present model are that the normal stress parallel to crack faces is included in the yielding condition in the cohesive zone and traction force in the cohesive zone can be non‐uniform. For a ductile material defined by von Mises criterion without hardening, the present model confirms that the normal stress parallel to crack face increases with increasing crack speed and can be even larger than the normal traction in the cohesive zone, which justifies the necessity of including the normal stress parallel to the crack faces in the yielding condition at high crack speed. In addition, strain hardening effect is examined based on a non‐uniform traction distribution in the cohesive zone.     

混凝土断裂过程区长度计算方法研究

该文基于粘聚裂缝概念,以起裂韧度作为裂缝起裂和扩展的准则,提出了混凝土断裂过程区长度的计算方法。以Ⅰ型裂缝为例,计算了不同初始缝长和起裂韧度情况下的断裂过程区长度值,结合以往大体积混凝土的试验数据对其进行了验证。进而分析了断裂过程区长度的影响因素,结果表明:断裂过程区长度随初始缝长的增大而逐渐增大,随起裂韧度的增大而逐渐减小。